nPr and nCr — arrangements and selections from a set.

What's the difference between permutations and combinations?

Permutations count arrangements where order matters — 1st, 2nd, and 3rd place among 5 runners is a permutation, since swapping who's 1st and who's 2nd creates a different outcome. Combinations count selections where order doesn't matter — choosing any 3 of those 5 runners for a relay team is a combination, since the same 3 people chosen in any order count as one outcome.

What does n! (factorial) mean?

n! (read "n factorial") is the product of all positive integers up to n — 5! = 5×4×3×2×1 = 120. Both the permutation and combination formulas are built from factorials, though this tool computes them via a running product to avoid overflow on larger numbers.

Why does nCr equal nPr divided by r!?

Permutations count every ordering of a chosen group as a separate outcome, but combinations treat all r! possible orderings of the same group as a single outcome. Dividing nPr by r! removes exactly those duplicate orderings, leaving just the count of distinct groups.

What's a real-world example of each?

Permutations: arranging books in a specific order on a shelf, or assigning gold/silver/bronze medals. Combinations: choosing a committee of people from a larger group, or picking lottery numbers, where the order you select them doesn't change the outcome.

Worked example: for n=10, r=3: permutations (order matters) = 10×9×8 = 720. Combinations (order doesn't matter) = 720/3! = 720/6 = 120 — exactly 6 times fewer, since each group of 3 can be arranged in 3! = 6 different orders.

Order is the whole distinction

A permutation counts arrangements where order matters; a combination counts selections where it doesn't. From 5 items choosing 3, there are P(5,3) = 60 permutations but only C(5,3) = 10 combinations — exactly 6 times fewer, because each group of 3 can be ordered in 3! = 6 ways.

That relationship always holds: C(n,r) = P(n,r) ÷ r!. If you are unsure which you need, ask whether swapping two chosen items produces a different outcome. Gold and silver medals: yes, that's a permutation. Three people on a committee: no, that's a combination.

Worked distinctions

• A 4-digit PIN — order matters and repetition is allowed, so it's 10⁴ = 10,000, not a permutation at all.
• Finishing positions in a race — permutation.
• A lottery draw — combination; the balls' order of appearance is irrelevant.
• Seating people around a table — a circular permutation, which is (n−1)! rather than n!, because rotating everyone produces the same arrangement.

With and without repetition

The standard formulas assume each item is used at most once. If repetition is allowed the arithmetic changes completely: arrangements become n^r, and selections become C(n+r−1, r). A combination lock allowing repeated digits is a different calculation from one that doesn't — and, despite the name, a combination lock is a permutation.

Why the numbers explode

Factorial growth is faster than exponential growth, and much faster than people expect. 10! is 3,628,800; 20! is over 2.4 × 10¹⁸. This is why exhaustive search becomes impossible so abruptly in scheduling and routing problems — adding one more item multiplies the work rather than adding to it.

For counting problems that involve probability rather than raw arrangements, pair this with the Statistics Calculator.